Finite Field

High Quality Content by WIKIPEDIA articles! In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, coding theory and quantum error correction. The finite fields are classified by size; there is exactly one finite field up to isomorphism of size pk for each prime p and positive integer k. Each finite field of size q is the splitting field of the polynomial xq − x, and thus the fixed field of the Frobenius endomorphism which takes x to xq. Similarly, the multiplicative group of the field is a cyclic group. Wedderburn\'s little theorem states that the Brauer group of a finite field is trivial, so that every finite division ring is a finite field. Finite fields have applications in many areas of mathematics and computer science, including coding theory, LFSRs, modular representation theory, and the groups of Lie type. Finite fields are an active area of research, including recent results on the Kakeya conjecture and open problems on the size of the smallest primitive root. Dannoe izdanie predstavlyaet soboj kompilyatsiyu svedenij, nahodyaschihsya v svobodnom dostupe v srede Internet v tselom, i v informatsionnom setevom resurse "Vikipediya" v chastnosti. Sobrannaya po chastotnym zaprosam ukazannoj tematiki, dannaya kompilyatsiya postroena po printsipu podbora blizkih informatsionnyh ssylok, ne imeet samostoyatelnogo syuzheta, ne soderzhit nikakih analiticheskih materialov, vyvodov, otsenok moralnogo, eticheskogo, politicheskogo, religioznogo i mirovozzrencheskogo haraktera v otnoshenii glavnoj tematiki, predstavlyaya soboj isklyuchitelno faktologicheskij material. This book was created using print-on-demand technology.